Theorem sbcalfi 39142

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 39140	. 2 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝐴 / 𝑥]𝜑)
3		sbcalfi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
4	3	albii 1851	. 2 ⊢ (∀𝑦[𝐴 / 𝑥]𝜑 ↔ ∀𝑦𝜓)
5	2, 4	bitri 278	1 ⊢ ([𝐴 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦𝜓)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  Ⅎwnfc 2907  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-v 3452  df-sbc 3739

This theorem is used by: (None)