Theorem 2stdpc4 2070

Description: A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2068 for the analogous single specialization. See 2sp 2186 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)

Assertion

Ref	Expression
2stdpc4	⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Proof of Theorem 2stdpc4

Step	Hyp	Ref	Expression
1		stdpc4 2068	. . 3 ⊢ (∀𝑦𝜑 → [𝑤 / 𝑦]𝜑)
2	1	alimi 1811	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥[𝑤 / 𝑦]𝜑)
3		stdpc4 2068	. 2 ⊢ (∀𝑥[𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4	2, 3	syl 17	1 ⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Syntax hints:   → wi 4  ∀wal 1538  [wsb 2064

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

This theorem depends on definitions:  df-bi 207  df-sb 2065

This theorem is referenced by:  ax11-pm2  36837  pm11.11  44393