Theorem sbcalfi 38117

Description: Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)

Hypotheses

Ref	Expression
sbcalfi.1	⊢ Ⅎ𝑦𝐴
sbcalfi.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbcalfi	⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbcalfi

Step	Hyp	Ref	Expression
1		sbcalfi.1	. . 3 ⊢ Ⅎ𝑦𝐴
2	1	sbcalf 38115	. 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
3		sbcalfi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
4	3	albii 1818	. 2 ⊢ (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓)
5	2, 4	bitri 275	1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   ↔ wb 206  ∀wal 1537  Ⅎwnfc 2890  [wsbc 3794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3483  df-sbc 3795

This theorem is referenced by: (None)