Theorem sbcexfi 38108

Description: Move existential 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
sbcexfi.1	⊢ Ⅎ𝑦𝐴
sbcexfi.2	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)

Assertion

Ref	Expression
sbcexfi	⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcexfi

Step	Hyp	Ref	Expression
1		sbcexfi.1	. . 3 ⊢ Ⅎ𝑦𝐴
2	1	sbcexf 38106	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3		sbcexfi.2	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜓)
4	3	exbii 1848	. 2 ⊢ (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
5	2, 4	bitri 275	1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   ↔ wb 206  ∃wex 1779  Ⅎwnfc 2878  [wsbc 3761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-v 3457  df-sbc 3762

This theorem is referenced by: (None)