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Theorem rspsbc2 39236
Description: rspsbc 3713 with two quantifying variables. This proof is rspsbc2VD 39578 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rspsbc2 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rspsbc2
StepHypRef Expression
1 idd 24 . 2 (𝐴𝐵 → (𝐶𝐷𝐶𝐷))
2 rspsbc 3713 . . . 4 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐴 / 𝑥]𝑦𝐷 𝜑))
32a1d 25 . . 3 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐴 / 𝑥]𝑦𝐷 𝜑)))
4 sbcralg 3708 . . . 4 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
54biimpd 220 . . 3 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 → ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
63, 5syl6d 75 . 2 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑 → ∀𝑦𝐷 [𝐴 / 𝑥]𝜑)))
7 rspsbc 3713 . 2 (𝐶𝐷 → (∀𝑦𝐷 [𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
81, 6, 7syl10 79 1 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2156  wral 3096  [wsbc 3633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-v 3393  df-sbc 3634
This theorem is referenced by:  tratrb  39238  tratrbVD  39585
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